§2. Series expansions
383
(14.31)
zn - 1 = II (z - e 2ki7t "/n) .
O::;k::;n-l
More generally,
(14.32)
for any nonzero complex number a = JaJe it , where JaJl/n is the usual positive
root. In these formulae, k E Z can vary "modulo n". On replacing z by x/y
in (31), one finds the identity
(14.33)
xn - yn = (x - y)(x - wy) ... (x - wn-1y)
where w = exp(21fi/n).
15 - Euler's relation chez Euler
In his Introductio in Analysin Infinitorum of 1748 which served as a Bible
to mathematicians up at least to Cauchy's time, Euler did not establish the
relation exp(ix) = cosx+isinx by means of power series. He deduced it from
de Moivre's formula (which did not presuppose these series) by an argument
as always very ingenious and, as always, a little false. He wrote
cos x + isinx = [cos(x/n) + isin(x/n)t
and made n tend to infinity. Then "evidently" cos(x/n) = 1 and sin(x/n) =
x/n, so that the right hand side is "clearly" equal to (1 + ix/n)n, "qed", if
we know that e Z = lim(1 + z/n)n.
To justify this argument, we have to put
cos(x/n) + i sin(x/n) = 1 + zn/n
and use Theorem 10: essentially, we have to show that Zn tends to ix, in other
words that
lim{n[cos(x/n) - 1] + insin(x/n)} = ix.
Since cos(x/n) - 1 lies between -x 2 /2n 2 and 0 for n large, by (14.9), the
product over n tends to o. It remains to prove that nsin(x/n) tends to x, i.e.
that the derivative of t ---+ sin t is equal to 1 for t = o. Qed, but (14.9) rests
on the series for cosine.
The interest of (14.28) and of the analogous formula deduced from it on
replacing i by -i (or, what comes to the same, x by -x) is to provide the no
less famous relations
(15.1')
cosnx
cos n X - ( ; ) cosn- 2 x. sin 2 x +
+ ( n) n-4 . 4
4 cos
x. sm x - ...